After going through the invisible staircase in 'Connecting the Pieces,' you come to a room with a bunch of spinning numbers, and some math symbols (+, =) on the floor. The numbers change depending on what angle you view them from, and the room hint is "Peeking behind the curtains lets us see how everything works."

It seems like I have to make the equation valid so I can go through the blocked off hallway the arrow is pointing to, but I spent about half an hour trying various things with no luck. So, what do I do in this room?

It looks like it's not enough to make the equation valid. If you stand on the equals sign, you can see 1+4=5, but nothing happens.
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murgatroid99Feb 7 '13 at 7:19

I'm not sure there is a puzzle here. There are several of these Peeking behind the curtains rooms. The point of them is to show you how the game works. I think this particular room is supposed to demonstrate how the game changes things around you based on where you are and where you look.
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GieronFeb 7 '13 at 8:24

@Gieron: But then, what's with the math symbols/arrow on the ground? And why can you see a staircase behind the glass (where the arrow is pointing) that you can't get to?
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BlueRaja - Danny PflughoeftFeb 7 '13 at 8:28

1

@BlueRaja, both answers are correct but you did make me curious about the staircase. Turns out it's the one that connects "Hidden Secrets" with the left doorway from "Stop and Go". I can post screenshot proof if you have trouble verifying it yourself.
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user42252Feb 11 '13 at 10:57

Well then I'm just curious what it's supposed to mean (or why there's an arrow pointing to a yellow staircase we can't get to...). I have a feeling it is (or was) supposed to be a puzzle, but is bugged or never finished.
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BlueRaja - Danny PflughoeftFeb 11 '13 at 6:20

That staircase might be a part of an entirely different section of the game. Remember: just because you can see the staircase from the number-box room doesn't necessarily mean that you can see the number-box room from the staircase. Regarding the meaning of the room: I think Shaamaan explains pretty well what the author wants to demonstrate.
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PhilippFeb 11 '13 at 12:53

It's a peak behind the curtains, so to speak, to show you some of the magic that happens in the game. My guess is that it's to show you how the 4D art gallery works - that it combines different states depending on where you stand. I also think the output cube is the one you can walk around without seeing it "twitch" and change states while you're clearly looking at one side.

There are 4 cubes inside the room. One of said cubes (the one closest to the entrance), has transparent sides. In the screenshot below all 4 cubes are visible. Also, since it'll be important later on, lets define 2 axis in this screenshot: X (moving left-right) and Y (moving towards the back wall and towards the entrance). This will be important later on.

The cube on the left contains 2 states - it will show a 1 or a 3, depending on where the player is positioned. Since this room is a "look how it works" kind of room, the player can see the cube change states.

The cube(s) in the middle contain 2 states. It will show either a 2 or a 4 depending on the players position. Like before, the player can see the states being changed. The cube with the transparent side already shows a "half-finished" product (thus the arrow on the floor), as the player will never witness this cube "twitch" between states because the player can only see one numbered-side at a time.

Note, that the 1-3 cube will change states when the player moves on a different axis than the 2-4 cube! Ie. the 1-3 cube will change states when the player moves along the X axis, and the 2-4 cube will change states when the player moves along the Y axis.

Below is the final, output cube. The player will never see it "twitch" (like the other cubes in the room), as it's the final product, a sum, so to speak, of the other cubes in the room and their states. In other words, the sides of this cube change, but the player never witnesses it, because it only occurs to sides that aren't visible.

PS. As it is, the puzzles (and the game in general) make a LOT of sense. Granted, some of them were more abstract than others, but they all fit well together. There are number of "peek behind the curtains" rooms in the game, showing how things work

like a room where you can see the optimization mechanics to reduce the number of polygons when cubes are placed

or showing how things were planned (during development) or are planned in the final version

like the room showing the layout for Finding the Seams maze.

If this DOES turn out to be a puzzle room, then it would be a really poor exception, and I think I'll need to change my opinion of the game. >_<'

There's a lot of things like that in the game. There's even one area with concept art, and another one with various different work-in-progress models of your gun.
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TacroyFeb 7 '13 at 17:16

But then what's with the staircase you can't get to that the arrow is pointing at? And the math symbols on the ground?
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BlueRaja - Danny PflughoeftFeb 7 '13 at 20:42

If I'm right and this room is there to show you how the 4D art gallery works, then you're thinking of the signs at the floor wrong. The idea is to produce a cube which has a different 3D object inside it depending on the side you're looking at. As such, the numbers (1-4) aren't meant to add up as numbers, but as sides to a final cube which displays all of them depending on where you stand.
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ShaamaanFeb 8 '13 at 7:24

@Shaamaan: That's not what it does, though. The cube on the right (which you can only see the edge of in my screenshot) never changes, though every side looks different. The cube in front of and behind the arrow are always the same, and switch between 2 and 4 depending on if you're on the left or the right of the room; and the one next to the '+' changes between 1 and 3 when you move from the front or back of the room.
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BlueRaja - Danny PflughoeftFeb 8 '13 at 8:36

The cube on the right (in your screenshot) isn't the final cube, I think. The final one is "behind" you in the screenshot - it's the 3D (4D?) outcome that's the "sum" of the previous cubes, which IMHO is a combination of the other two cubes and their states. For the math part, take note: there are numbers from 1-4. You can have both 1 and 4 visible, but there's no 5 anywhere, so the idea of a mathematical sum goes right out the window (well, the rotating 2 may look like a 5, but then you'd need 5 different colour backgrounds).
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ShaamaanFeb 9 '13 at 12:02

The room explains how the game renders cubes that have different contents depending on your angle of view (like those that appear in the gallery). I'll describe what each cube does. Front/rear and left/right will refer to the point of view of someone just entering the room.

The rightmost cube, next to the equal sign, is the rendered output: a cube that shows a different number through every side. That's not how real-world optics work, and game engines won't handle the non-standard geometry directly. So the end product is rendered by combining two offscreen scenes, that individually have standard 3D geometry.

The left cube is an input that will be rendered offscreen, it contains side 1 and side 3 of the output. Since 1 and 3 are opposite sides and won't be visible to the player at the same time, the cube doesn't need to render both, and switches depending on the position of the player.

Sides 2 and 4 are rendered from the middle cube, which is shown unoptimised before the arrow and optimised after the arrow. Faces 1 and 3 (front and rear of the room) aren't rendered in the optimised cube, since they'll be taken from the left cube.

And that's how you obtain the final cube: render two scenes (with occlusion masks that tell the engine which parts can be optimised out) and composite them offscreen. A more complicated scene like the gallery still needs only two offscreen scenes, but will have a lot more triggers to switch states between opposite faces (one scene will contain a front/rear trigger for every cube, the other will contain left/right triggers). It explains the flat, abstract look of the game: if the game rendered shadows, lights and reflections, the composited output would contain impossible lighting/shadows (from invisible/offscreen objects) and immersion would suffer.